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Discovering Advanced Algebra An Investigative Approach 1st edition Jerald Murdock, Ellen Kamischke, Eric Kamischke - Solutions

An object is projected upward, and these data are collected.a. Write a function that relates time and height for this object. b. What was the initial height? The initial velocity? c. When does the object reach its maximum height? What is the maximum height?
The members of the Young Entrepreneurs Club decide to sell T-shirts in their school colors for Spirit Week. In a marketing survey, the members ask students whether or not they would buy a T-shirt for a specific price. In analyzing the data, club members find that at a price of $20 they would sell
Multiply. a. (x - 3) (2x + 4) b. (x2 + 1) (x + 2)
Solve (x - 2) (x + 3)(2x - 1) = 0.
Consider a graph of the unit circle, x2 + y2 = 1. Stretch it vertically by a scale factor of 3, and translate it left 5 units and up 7 units. a. Write the equation of this new shape. What is it called? b. Sketch a graph of this shape. Label the center and at least four points.
This table shows the number of endangered species in the United States for selected years from 1980 to 2000.a. Define variables and create a scatter plot of these data. b. Find the equation of the median-median line for these data. c. Use the median-median line to predict the number of endangered
What value is required to complete the square?
Convert each quadratic function to vertex form. a. y = x2 + 20x + 94 b. y = x2 - 7x + 16 c. y = 6x2 - 24x + 147 d. y = 5x2 + 8x
Rewrite each expression in the form (x2 + bx + c, and then identify the coefficients, (, b, and c. a. 3x2 + 2x - 5 b. 14 + 2x2 c. -3 + 4x2 - 2x + 8x d. 3x - x2
What is the vertex of the graph of the quadratic function y = -2x2 - 16x - 20?
Convert the function y = 7.51x2 - 47.32x + 129.47 to vertex form. Use a graph or table to verify that the functions are equivalent.
Imagine that an arrow is shot from the bottom of a well. It passes ground level at 1.1 s and lands on the ground at 4.7 s.a. Define variables and write a quadratic function that describes the height of the arrow, in meters, as a function of time.b. What was the initial velocity of the arrow in
Suppose you are enclosing a rectangular area to create a rabbit cage. You have 80 ft of fence and want to build a pen with the largest possible area for your rabbit, so you build the cage using an existing building as one side.a. Make a table showing the areas for some selected values of x, and
A rock is thrown upward from the edge of a 50 m cliff overlooking Lake Superior, with an initial velocity of 17.2 m/s. Define variables and write an equation that models the height of the rock.
Solve. a. (x - 2.3)2 = 25 b. (x + 4.45)2 = 12.25 c. (x - 3/4)2 = 25 / 16
Show that the mean of the two solutions provided by the quadratic formula is - b/2(. Explain what this tells you about a graph.
These data give the amount of water in a draining bathtub and the amount of time after the plug was pulled.a. Write a function that gives the amount of water as a function of time. b. How much water was in the tub when the plug was pulled? c. How long did it take the tub to empty?
A golden rectangle is a rectangle that can be divided into a square and another rectangle that is also a golden rectangle similar to the original. In the figure at right, ABCD is a golden rectangle because it can be divided into square ABFE and golden rectangle FCDE. Setting up a proportion of the
Complete each equation.
Find the inverse of each function. (The inverse does not need to be a function.)a. y = (x + 1)2b. y = (x + 1)2 + 4c. y = x2 + 2x - 5
Convert these quadratic functions to general form. a. y = (x - 3)(2x + 5) b. y = -2(x - 1)2 + 4
A 20 ft ladder leans against a building. Let x represent the distance between the building and the foot of the ladder, and let y represent the height the ladder reaches on the building.a. Write an equation for y in terms of x.b. Find the height the ladder reaches on the building if the foot of the
The main cables of a suspension bridge typically hang in the shape of parallel parabolas on both sides of the roadway. The vertical support cables, labeled a-k, are equally spaced, and the center of the parabolic cable touches the roadway at f. If this bridge has a span of 160 ft between towers,
Rewrite each equation in general form, ax2 + bx + c = 0. Identify (, b, and c. a. 3x2 - 13x = 10 b. x2 - 13 = 5x c. 3x2 + 5x = -1 d. 3x2 - 2 - 3x = 0 e. 14(x - 4) - (x + 2) = (x + 2)(x - 4)
Evaluate each expression using your calculator. Round your answers to the nearest thousandth.a.b. c. d.
Solve by any method. a. x2 - 6x + 5 = 0 b. x2 - 7x - 18 = 0 c. 5x2 + 12x + 7= 0
Use the roots of the equations in Exercise 4 to write each of these functions in factored form, y = a(x - r1) (x - r2). a. y = x2 - 6x + 5 b. y = x2 - 7x - 18 c. y = 5x2 + 12x + 7
Beth uses the quadratic formula to solve an equation and getsa. Write the quadratic equation Beth started with. b. Write the simplified forms of the exact answers. c. What are the x-intercepts of the graph of this quadratic function?
Write a quadratic function whose graph has these x-intercepts. a. 3 and -3 b. 4 and - 2/5 c. r1 and r2
Use the quadratic formula to find the zeros of y = 2x2 + 2x + 5. Explain what happens. Graph y = 2x2 + 2x + 5 to confirm your observation. How can you recognize this situation before using the quadratic formula?
Add or subtract. a. a. (5 - 1i) + (3 + 5i) b. (6 + 2i) - (-1 + 2i) c. (2 + 3i) + (2 - 5i) d. (2.35 + 2.71i) - (4.91 + 3.32i)
Solve each equation. Label each solution as real, imaginary, and/or complex.a. x2 - 4x + 6 = 0b. x2 + 1 = 0c. x2 + x = -1d. x2 - 1 = 0
Write a quadratic function in general form that has the given zeros, and leading coefficient of 1. a. x = -3 and x = 5 b. x = -3.5 (double root) c. x = 5i and x = -5i d. x = 2 + i and x = 2 - i
Write a quadratic function in general form that has a zero of x = 4 + 3i and whose graph has y-intercept 50.
Solve.a. x2 - 10ix - 9i2 = 0b. x2 - 3ix = 2
The quadratic formula,
Use these recursive formulas to find the first six terms (z0 to z5) of each sequence. Describe what happens in the long run for each sequence. a. z0 = 0 zn = zn - 12 + 0 where n ( 1 b. z0 = 0 zn = zn - 12 + i where n ( 1 c. z0 = 0 zn = zn - 12 + 1 - i where n ( 1 d. z0 = 0 zn = zn - 12 + 0.2 + 0.2i
Consider the function y = 2x2 + 6x - 3. a. List the zeros in exact radical form and as approximations to the nearest hundredth. b. Graph the function and label the exact coordinates of the vertex and points where the graph crosses the x-axis and the y-axis.
Consider two positive integers that meet these conditions: three times the first added to four times the second is less than 30 twice the first is less than five more than the second a. Define variables and write a system of linear inequalities that represents this situation. b. Graph the feasible
Multiply.a. (5 - 1i) (3 + 5i)b. 6 (-1 + 2i)c. 3i (2 - 5i)d. (2.35 + 2.71i) (4.91 + 3.32i)
Find the conjugate of each complex number. a. 5 - i b. -1 + 2i c. 2 + 3i d. -2.35 - 2.71i
Draw Venn diagrams to show the relationships between these sets of numbers. a. Real numbers and complex numbers b. Rational numbers and irrational numbers c. Imaginary numbers and complex numbers d. Imaginary numbers and real numbers e. Complex numbers, real numbers, and imaginary numbers
Rewrite this quadratic equation in general form. (x - (2 + i))(x - (2 - i)) = 0
Use the definitions i = (-1 and i2 = -1 to rewrite each power of i as 1, i, -1, or -i. a. i3 b. i4 c. i5 d. i10
Plot the numbers i, i2, i3, i4, i5, i6, i7, and i8 on a complex plane. What pattern do you see? What do you expect the value of i17 to be? Verify your conjecture.
Rewrite the quotient 2 + 3i / 2 - i in the form ( + bi.
Without graphing, find the x-intercepts and the y-intercept for the graph of each equation. Check each answer by graphing. a. y = -0.25(x + 1.5)(x + 6) b. y = 3(x - 4)(x - 4) c. y = -2(x - 3)(x + 2)(x + 5) d. y = 5(x + 3)(x + 3)(x - 3)
Find the quadratic function whose graph has vertex (-2, 3) and contains the point (4, 12).
Find all real solutions.a. x2 = 50.4b. x4 = 169c. (x - 2.4)2 = 40.2d. x3 = -64
Algebraically find the inverse of each function. Then choose a value of x and check your findings. a. ((x) = 2/3 (x + 5) b. g(x) = - 6 + (x + 3)2/3 c. h(x) = 7 - 2x
Use the finite differences method to find the function that generates this table of values. Explain your reasoning.
Write the factored form of the quadratic function for each graph. Don't forget the vertical scale factor.a.b.
Convert each polynomial function to general form. a. y = (x - 4)(x - 6) b. y = (x - 3)(x - 3) c. y = x(x + 8)(x - 8) d. y = 3(x + 2)(x - 2)(x + 5)
Given the function y = 2.5(x - 7.5)(x + 2.5)(x - 3.2),a. Find the x-intercepts without graphing. b. Find the y-intercept without graphing.c. Write the function in general form.d. Graph both the factored form and the general form of the function to check your work.
Use your work from the investigation to answer these questions. a. What x-value maximizes the volume for your box? What is the maximum volume possible? b. What x-value or values give a volume of 300 cubic units? c. The portion of the graph with domain x > 10 shows positive volume. What does this
Write each polynomial as a product of factors.a. 4x2 - 88x + 480b. 6x2 - 7x - 5c. x3 + 5x2 - 4x - 20d. 2x3 + 16x2 + 38x + 24e. a2 + 2ab + b2f. x2 - 64g. x2 + 64h. x2 - 7i. x2 - 3x
Sketch a graph for each situation if possible. a. A quadratic function with only one real zero b. A quadratic function with no real zeros c. A quadratic function with three real zeros d. A cubic function with only one real zero e. A cubic function with two real zeros f. A cubic function with no
Consider the function in this graph.a. Write the equation of a polynomial function that has the x-intercepts shown in the graph.Use a for the vertical scale factor.b. Use the y-intercept to determine the vertical scale factor. Write the function from 8a, replacing a with the value of the vertical
The way you taste certain flavors is a genetic trait inherited from your parents. For instance, the ability to taste the bitter compound phenylthiocarbamide (PTC) is inherited as a dominant trait in humans. In the United States, approximately 70% of the population can taste PTC, whereas 30%
Identify the zeros of each function.a.b. c. d.
A 4th-degree polynomial function has the general form y = (x4 + bx3 + cx2 + dx + e for real values of a, b, c, d, and e, where a ( 0. Graph several 4th-degree polynomial functions by trying different values for each coefficient. Be sure to include positive, negative, and zero values. Make a sketch
Each of these is the graph of a polynomial function with leading coefficient ( = 1 or ( = -1.i.ii. iii. iv. a. Write a function in factored form that will produce each graph. b. Name the zeros of each polynomial function in 11a. If a factor is raised to the power of n, list the zero n times.
Consider the polynomial functions in Exercise 11.a. What is the degree of each polynomial function?b. How many extreme values does each graph have?c. What is the relationship between the degree of the polynomial function and the number of extreme values?d. Complete these statements:
In the lesson you saw various possible appearances of the graph of a 3rd-degree polynomial function, and in Exercise 10 you explored possible appearances of the graph of a 4th-degree polynomial function. In Exercises 11 and 12, you found a relationship between the degree of a polynomial function
Find the roots of these quadratic equations. Express them as fractions. a. 0 = 3x2 - 13x - 10 b. 0 = 6x2 - 11x + 3 c. List all the factors of the constant term, c, and the leading coefficient, a, for 14a and b. What do you notice about the relationship between the factors of a and c, and the roots
If 3 + 5(2 is a solution of a quadratic equation with rational coefficients, then what other number must also be a solution? Write an equation in general form that has these solutions.
Given the function (x) = x2 + 2x + 10, find these values.a. (-3)b. (-1/5)c. (2 - 3(2)d. (-1 + 3i)
Solve this system using each method specified.a. Use an inverse matrix and matrix multiplication. b. Write an augmented matrix, and reduce it to reduced row-echelon form.
According to Froude's Law, the speed at which an aquatic animal can swim is proportional to the square root of its length. (On Growth and Form, Sir D'Arcy Thompson, Cambridge University Press, 1961.) If a 75-foot blue whale can swim at a maximum speed of 20 knots, write a function that relates its
Give the coordinates of the y-intercept of each graph.a.b. c. d.
Identify the lowest possible degree of each polynomial function.a.b. c. d.
Write the factored form for each polynomial function. Check your work by graphing on your calculator.a.b. c. d.
Write polynomial functions with these features. a. A linear function whose graph has x-intercept 4. b. A quadratic function whose graph has only one x-intercept, 4. c. A cubic function whose graph has only one x-intercept, 4.
The graph of y = 2(x - 3)(x - 5)(x + 4)2 has x-intercepts 3, 5, and - 4 because they are the only possible x-values that make y = 0. This is a 4th-degree polynomial, but it has only three x-intercepts. The root x = - 4 is called a double root because the factor (x + 4) occurs twice. Make a complete
The graph at right is a complete graph of a polynomial function.a. How many x-intercepts are there?b. What is the lowest possible degree of this polynomial function?c. Write the factored form of this function if the graph includes the points (0, 0), (- 5, 0), (4, 0), (- 1, 0), and (1, 216).
Write the lowest-degree polynomial function that has the given set of zeros and whose graph has the given y-intercept. a. zeros: x = - 4, x = 5, x = - 2 (double root); y-intercept: - 80 b. zeros: x = - 4, x = 5, x = - 2 (double root); y-intercept: 160 c. zeros: x = 1/3, x = -2/5, x = 0;
Look back at Exercises 1-4. Find the products of the zeros in Exercise 1. How does the value of the leading coefficient, (, relate to the y-intercept, the product of the zeros, and the degree of the function?
Find the missing polynomial in each long-division problem.a.b.
When you trace the graph of a function on your calculator to find the value of an x-intercept, you often see the y-value jump from positive to negative when you pass over the zero. By using smaller windows, you can find increasingly more accurate approximations for x. This process can be automated
The relationship between the height and the diameter of a tree is approximately determined by the equation f (x) = kx3/2, where x is the height in feet, f (x) is the diameter in inches, and k is a constant that depends on the kind of tree you are measuring. a. A 221 ft British Columbian pine is
Find a polynomial function of lowest possible degree whose graph passes through the points (- 2, - 8.2), (- 1, 6.8), (0, 5), (1, - 1), (2, 1.4), and (3, 24.8).
Sam and Beth have started a hat business in their basement. They make baseball caps and sun hats. Let b represent the number of baseball caps, and let s represent the number of sun hats. Manufacturing demands and machinery constraints confine the production per day to the feasible region defined
Write each quadratic function in general form and in factored form. Identify the vertex, y-intercept, and x-intercepts of each parabola. a. y = (x - 2)2 - 16 b. y = 3(x + 1)2 - 27 c. y = -1/2 (x - 5)2 + 49/2 d. y = 2(x - 3)2 + 3
Solve. a. 6x + x2 + 5 = -4 + 4(x + 3) b. 7 = x(x + 3) c. 2x2 - 3x + 1 = x2 - x - 4
x3 + 2x2 + 3x - 6 = (x - 1)(x2 + 3x + 6).
Find the missing value in each synthetic-division problem.a.b. c. d.
Use the dividend, divisor, and quotient to rewrite each synthetic-division problem in Exercise 3 as a factored product in the form P(x) = D(x) · (x).
Make a list of the possible rational roots of 0 = 2x3 + 3x2 - 32x + 15.
a. P(x) = 47, D(x) = 11b. P(x) = 6x4 - 5x3 + 7x2 - 12x + 15, D(x) = x - 1c. P(x) = x3 - x2 - 10x + 16, D(x) = x - 2
Consider the function P(x) = 2x3 - x2 + 18x - 9. a. Verify that 3i is a zero. b. Find the remaining zeros of the function P.
Consider the function y = x4 + 3x3 - 11x2 - 3x + 10. a. How many zeros does this function have? b. Name the zeros. c. Write the polynomial function in factored form.
Use your list of possible rational roots from Exercise 5 to write this function in factored form. y = 2x3 + 3x2 - 32x + 15
Factor each expression completely. a. 2x2 - 10x + 12 b. 2x2 + 7x + 3 c. x3 - 10x2 - 24x
This 26-by-21 cm rectangle has been divided into two regions. The width of the unshaded region is x cm, as shown.a. Express the area of the shaded part as a function of x, and graph it. b. Find the domain and range for this function. c. Find the x-value that makes the two regions (shaded and
Consider the polynomial equation 0 = 3x4 - 20x3 + 68x2 - 92x - 39. a. List all possible rational roots. b. Find the four roots of the equation.
Write each expression in the form a + bi. a. (4 - 2i) (-3 + 6i) b. (-3 + 4i) - (3 + 13i) c. 2 - i / 3 - 4i
Divide.
Solve each equation by setting it equal to zero and factoring. a. x2 - 8x = 9 b. x4 + 2x3 = 15x2
Using three noncollinear points as vertices, how many different triangles can you draw? Given a choice of four points, no three of which are collinear, how many different triangles can you draw? Given a choice of five points? n points?
Tell whether each equation is written in general form, vertex form, or factored form. Write each equation in the other two forms, if possible. a. y = 2(x - 2)2 - 16 b. y = -3(x - 5)(x + 1) c. y = x2 + 3x + 2 d. y = (x + 1)(x - 3)(x + 4) e. y = 2x2 + 5x - 6 f. y = -2 - (x + 7)2

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